nth term divergence test | Series | AP Calculus BC | Khan Academy | Summary and Q&A

TL;DR

The divergence test is used to determine if an infinite series will definitely diverge based on the limit as N approaches infinity, and it cannot determine if a series will converge.

Key Insights

• 🏆 The divergence test is a simple and intuitive method for determining if an infinite series will diverge.
• 🙅 The test relies on the limit as N approaches infinity to assess the behavior of the terms in the series.
• ⛔ If the limit is not equal to zero, the series will definitely diverge.
• 🏆 The divergence test cannot determine if a series will converge, as there are other tests available for that purpose.

Transcript

• [Voiceover] What we're going to do now is start to explore a series of tests to determine whether a series will converge or diverge and the first one I'm going to go through right now is perhaps the most basic and hopefully see the most intuitive and this is the divergence test. The divergence test won't tell us if a series will converge but it c... Read More

Questions & Answers

Q: How does the divergence test help determine if an infinite series will converge or diverge?

The divergence test helps determine if an infinite series will definitely diverge by checking if the limit as N approaches infinity of A sub N is not equal to zero. If it is not zero, the series will diverge.

Q: Can the divergence test determine if a series will converge?

No, the divergence test can only determine if a series will definitely diverge. It cannot determine if a series will converge.

Q: What is the significance of the limit as N approaches infinity in the divergence test?

The limit as N approaches infinity helps determine the behavior of the terms in the infinite series. If the limit is not zero, it indicates that the series will diverge.

Q: Why is it important for the terms in an infinite series to approach zero for convergence?

In order for an infinite series to converge, the terms must get smaller and smaller as N approaches infinity. If the terms do not approach zero, the series will diverge.

Summary & Key Takeaways

• The divergence test is a basic and intuitive method to determine if an infinite series will diverge or not.

• The test states that if the limit as N approaches infinity of A sub N does not equal zero, then the series will diverge.

• The test is useful for identifying series that definitely diverge, but it cannot determine if a series will converge.